23

間接フィードバック制御による振り子の振動制御
園部 元康＊・東 正樹＊＊

Vibration Control of a Pendulum
by Indirect Feedback Control
Motomichi Sonobe＊, Masaki Higashi＊＊
Abstract
There are some approaches to control the vibration of a pendulum. Our control model is composedof a pendulum and cart in a cart-type single pendulum system. However, our control method suggested
in this report removes the pendulum from control model, and give the reference motions to state variablesof the cart in order to control the vibration of pendulum. The reference motions are defined by Jacobian
elliptic function which shows the solution of the free vibration of a pendulum. Consequently, our control
system regards vibration control of pendulum as a subsystem which is indirect feedback system. In thisreport, our control method is formulated and the effectiveness is verified by numerical simulation.

1 . ॹ‫ݴ‬ΫϨʔϯʹ୅ද͞ΕΔৼΓࢠ‫ߏܕ‬଄෺ͷৼಈ੍‫ޚ‬Λߦ͏ࡍͷ୅දతͳ੍‫ޚ‬ख๏ͱͯ͠ɼৼΓࢠΛ࠷Լ఺෇ۙͰۙࣅઢ‫ܗ‬ԽΛߦͬͨγεςϜʹରͯ͠ઢ‫ޚ੍ܗ‬Λ
[1] [2]ߦ͏ख๏΍ɼΦʔϓϯϧʔϓͰঢ়ଶྔΛ‫ܭ‬ଌ͢Δ͜ͱͳ͘ৼΓࢠͷৼಈΛখ͘͞཈͑Δ੍‫يޚ‬ಓΛੜ
[3] [4]੒͢Δख๏͕ҰൠతͰ͋Δɽಛʹલऀ͸ɼ࠷ద੍‫ޚ‬ཧ࿦Λద༻֤͠ঢ়ଶྔͷऩଋ͓Αͼ੍‫ྔޚ‬ͷେ͖͞ʹ߹Θ੍ͤͨ‫ܥޚ‬ઃ‫ܭ‬Λߦ͏͜ͱ͕Ͱ͖Δɽ͔͠͠ɼৼΓࢠͷৼ෯͕େ͖͘ɼඇઢ‫ܗ‬ಛੑ͕‫͢༻࡞͘ڧ‬Δ৔߹ʹ͸ۙࣅઢ‫ܗ‬Խख๏͸ͷద༻͸೉͘͠ͳΔɽඇઢ‫ܗ‬ಛੑʹର͢Δରࡦͱͯ͠͸ɼ෦෼ઢ‫ܗ‬Խ੍‫ޚ‬΍Τωϧ[5]Ϊʔ੍‫ޚ‬๏ ͳͲͷඇઢ‫ޚ੍ܗ‬ख๏͕͋Δɽ͜ΕΒͷ੍‫ޚ‬ख๏ͷ‫ڞ‬௨͢Δͷ͸ɼͦͷ੍‫ޚ‬ੑೳ͸ઃ‫͕ऀܭ‬༩͑Δ‫ۃ‬഑ஔ͋Δ͍͸ධՁؔ਺ͷॏΈʹґଘ͢Δͱ͍͏͜ͱͰ͋Δɽ͜Εʹରͯ͠ɼຊ‫ڀݚ‬͸‫ۃ‬഑ஔ΍ධՁؔ਺Ͱ͸ͳ͘ɼ‫ܥ‬ͷྗֶಛੑʹै੍ͬͯ‫ޚ‬Λߦ͏ख๏ʹ͍ͭͯ‫ݕ‬౼͢Δ΋ͷͰ͋ΔɽৼΓࢠͷࣗ༝ৼಈ͸ɼৼΓࢠͷॏ͞ɼ௕͞ͳͲͷ‫ܥ‬ύϥϝʔλͳͲʹΑͬͯྗֶಛੑ͕ܾ·͍ͬͯΔɽͦͷಛੑʹ߹Θ੍ͤͨ‫ޚ‬Λߦ͏͜ͱ͕Ͱ͖Ε͹ɼ͔ࣗͣΒͦͷऩଋੑͳͲͷ੍‫ޚ‬ੑೳ͕ܾ·ͬͯ͘Δ͸ͣͰ͋ΓɼධՁؔ਺΍‫ʹۃ‬ґଘ͠ͳ੍͍‫ޚ‬ଇ͕ߏஙͰ͖Δͱߟ͑ΒΕΔɽৼΓࢠͷৼಈ੍‫ޚ‬Λߟ͑Δ্Ͱɼͦͷ੍‫ޚ‬໨తΛঢ়ଶྔͷ੍‫Ͱޚ‬͸ͳ͘ɼৼಈΛΑΓେ͖͘ʢ͋Δ͍͸ɼΑΓখ͘͞ʣ͢Δ͜ͱΛ໨ࢦ͢ɽͦͷࡍʹϒϥϯίͷ‫ݪ‬ཧͰ΋͋Δ‫ݱৼڞ‬৅ʹ‫͍ͯͮج‬ɼ‫ܥ‬ͷྗֶಛੑΛߟ੍ྀͨ͠‫ޚ‬ଇΛߟ͑Δɽ௨ৗͷઢ‫ܥܗ‬ͷ‫ݱৼڞ‬৅͸ɼ֎ྗͱͯ͠‫ܥ‬ͷ‫ݻ‬༗ৼಈ਺ͱಉ͡ৼಈ਺ͷࡾ֯ؔ਺Λ༩͑Δ͜ͱʹΑͬͯҾ͖‫͜͞ى‬ΕΔɽ͔͠͠ɼৼΓࢠͷ৔߹͸ඇઢ‫ܗ‬γεςϜͰ͋ΔͷͰɼࡾ֯ؔ਺ʹΑͬͯ‫ৼڞ‬ΛҾ͖‫͍ͳ͖Ͱ͕ͱ͜͢͜ى‬ɽຊ‫Ͱڀݚ‬͸ɼࡾ[6]֯ؔ਺ͷ୅ΘΓʹϠίϏͷପԁؔ਺ Λ༻͍Δ͜ͱͰɼ‫͍ۙʹৼڞ‬ঢ়ଶΛ࣮‫͢ݱ‬Δɽ·ͨɼҰൠతʹ͸‫ͱৼڞ‬͸ৼಈΛେ͖͘͢Δ‫ݱ‬৅Λࢦ͕͢ɼΞΫνϡΤʔλͱৼΓࢠͷҐ૬ࠩΛ‫ݻ‬ఆ͢Δ͜ͱʹΑͬͯɼ‫ৼʹٯ‬ಈΛখ͘͢͞Δ͜ͱ΋ՄೳͱͳΔɽҎ্ͷΑ͏ʹɼຊ‫Ͱڀݚ‬͸୆ंͱৼΓࢠͷؒʹඇઢ‫ݱৼڞܗ‬৅ʹྨࣅͨ͠ӡಈΛҾ͖‫ৼͰͱ͜͢͜ى‬Γࢠͷৼಈͷ੍‫ޚ‬Λߦ͏ɽͦͷࡍͷ੍‫ޚ‬γεςϜ͸୆ंͷΈΛ੍‫ޚ‬ର৅ͱ͢Δ੍‫ܥޚ‬Λߏங͠ɼৼΓࢠͷӡಈ͸୆ंͷӡಈͷ໨ඪ஋ʹӨ‫ڹ‬Λ‫઀ؒ͢΅ٴ‬ϑΟʔυόοΫ‫ܥ‬Λ࠾༻͢Δɽ͜ΕʹΑͬͯɼৼΓࢠͷৼಈ੍‫ޚ‬ͷΈͳΒͣ୆ंͷҐஔܾΊ੍‫ޚ‬౳ͱͷ૊Έ߹Θ͕ͤ༰қʹͳΔ͜ͱ͕‫ظ‬଴͞ΕΔɽ͜ΕΒͷఏҊ͢Δ੍‫ޚ‬ख๏Λɼ୆ं‫ܕ‬୯Ұ౗ཱৼࢠϞσϧͷৼಈ੍‫ʹޚ‬ద༻͠ɼ਺஋ɹˎ ిࢠ‫ػ‬ց޻ֶՊˎˎ ઐ߈Պੜ࢈γεςϜ޻ֶઐ߈

平成20年９月30日受理

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弓削商船高等専門学校 紀要 第31号（平成21年）

γϛϡϨʔγϣϯʹΑͬͯͦͷ༗ޮੑͷ‫ূݕ‬Λߦ͏ɽ

x2 . ৼΓࢠͷৼಈ੍‫ޚ‬ͷͨΊͷ୆ंӡಈl2. 1 ੍‫ޚ‬ର৅Ϟσϧ੍‫ޚ‬ର৅ϞσϧΛਤ 1 ʹࣔ͢ɽৼΓࢠͷӡಈ͸୆ंʹӨ‫ڹ‬Λ‫ͱ͍ͳ͞΅ٴ‬Ծఆ͢Δͱɼӡಈํఔࣜ͸࣍ࣜͰද͞ΕΔɽ

u(m, J)⎫⎪⎪⎬ẍ + 2ξζ ẋ = ξ 2 u− ml cos θ · ẍ + (J + ml2 )θ̈+ c θ̇ + mgl sin θ = 0

⎪
⎪
⎭

θ

(1)

Fig. 1 Single pendulum model͜͜ʹɼm ͸ৼΓࢠͷ࣭ྔɼl ͸ճస͔࣠Βॏ৺·Ͱͷ௕͞ɼJ ͸‫׳‬ੑϞʔϝϯτɼc ͸ৼΓࢠͷճసʹؔ͢Δ೪ੑ‫ݮ‬ਰ܎਺ɼu ͸੍‫ޚ‬ೖྗిѹɼg ͸ॏྗՃ଎l౓Ͱ͋Δɽ·ͨɼξ ͓Αͼ ζ ͸୆ंͷӡಈํఔࣜΛ؆қԽ͢ΔͨΊͷ෺ཧఆ਺Ͱ͋Δɽࣜ (1) ͷୈ 1 ࣜͷ؆қԽ͸࣮‫ػݧ‬ͷӡಈΛద੾ʹද‫͢ݱ‬ΔͨΊʹߦͬͨ΋ͷͰ͋Δɽ

(m, J)

ࣜ (1) ͷୈ 2 ࣜʹ͍ͭͯɼ୆ंͷՃ଎౓߲͕ৼΓࢠ΁ͷ‫ ྗ֎੍ڧ‬f ͱͯ͠࡞༻͍ͯ͠Δͱߟ͑Δͱɼ࣍ͷΑ͏ʹද͞ΕΔɽ

(J + ml2 )θ̈ + c θ̇ + mgl sin θ = f



f = ml cos θ · ẍθFig. 2 Simple pendulum model(2)͜͜ʹɼsn ͓Αͼ cn ͸ϠίϏͷପԁؔ਺ɼk ͸ପԁࣜ (2) ʹ͓͍ͯɼ‫ ྗ֎੍ڧ‬f ͕ৼΓࢠʹରͯ͠ਖ਼ͷ࢓ؔ਺ͷ฼਺Ͱ͋Δɽ͍·ɼ୆ंͷมҐ͕ৼΓࢠͷ֯଎ࣄΛߦ͏ͱৼΓࢠͷৼಈ͸খ͘͞ͳΓɼ‫ʹٯ‬ෛͷ࢓ࣄ౓ʹಉ‫ͯ͠ظ‬ಈ͍͍ͯΔͱԾఆ͢Δͱɼh Λൺྫఆ਺Λߦ͏͜ͱͰৼΓࢠͷৼಈΛେ͖͘͢Δ͜ͱ͕Ͱ͖Δɽͱͯ͠ x = hθ̇ ͱ͓͘͜ͱ͕Ͱ͖Δɽ͜ͷࣜΛ؆୯ʹପԁؔ਺ʹΑΔՃ଎౓ͷಋग़ͱ࢓ࣄ཰͢ΔͨΊʹɼh = a/2ωn k ͱͯࣜ͠ (4) ͷୈ 2 ࣜΛ୅

2. 2໨ඪͱ͢Δৼಈ੍‫ޚ‬Λ࣮‫͢ݱ‬ΔͨΊʹ͸ɼৼΓࢠͷӡಈͱಉ͡प‫ظ‬ͷӡಈΛ୆ंʹ༩͑Δඞཁ͕͋Δɽͦ͜Ͱɼ୆ंͷӡಈΛ༩͑Δؔ਺ΛৼΓࢠͷࣗ༝ৼಈղೖ͢Δͱɼ࣍ͷΑ͏ʹͳΔɽ
ax=θ̇ = a cn(ωn t, k)
2ωn k(5)͕ͳ͍ਤ 2 ʹࣔ͢୯ৼΓࢠϞσϧͷӡಈํఔࣜ͸࣍ͷcn ؔ ਺ ͸ cos ؔ ਺ ͱ ࣅ ͨ ੑ ࣭ Λ ࣋ ͪ ɼ−1 ≤cn (ωn t, k) ≤ 1 ͷ஋ҬͰԟ෮͢Δؔ਺Ͱ͋Δɽͨ͠
͕ͬͯɼࣜ (5) ͷ a ͸୆ंͷԟ෮ӡಈͷৼ෯ʹ૬౰͢Α͏ʹද͞ΕΔɽΔɽࣜ (5) Λ 2 ֊࣌ؒඍ෼ͯ͠୆ंͷՃ଎౓Λ‫ٻ‬ΊΔ͔Βಋग़͢Δɽ
ਤ 1 ͷϞσϧ͔Β୆ंΛআ͖ɼ͞ΒʹৼΓࢠͷ‫ݮ‬ਰmgl(3)J + ml2͜͜ʹɼωn ͸ৼΓࢠͷઢ‫ݻܗ‬༗֯ৼಈ਺Ͱ͋Δɽࣜ (3)ʹର͠ɼৼ෯Λ α rad ͱ͢ΔৼΓࢠͷෆ‫ݮ‬ਰࣗ༝ৼಈղ (֯มҐɼ֯଎౓ɼ֯Ճ଎౓) ͸ϠίϏͷପԁؔ਺Λθ̈ + ωn2 sin θ = 0,ωn =༻͍ͯ࣍ͷΑ͏ʹද͞ΕΔɽ⎫
θ = 2 sin−1 (k sn(ωn t, k))⎪⎪⎪⎪⎪⎪⎬
θ̇ = 2ω k cn(ω t, k)nθ̈ = −ωn2 sin θk = sin

α2n⎪
⎪⎪
⎪
⎪
⎪
⎭

(4)

ͱ࣍ͷΑ͏ʹͳΔɽ
aωnẍ = −θ̇ cos θ(6)2k୆ंͷӡಈ͕ࣜ (6) Λຬͨ͢ͱ͖ɼࣜ (2) ͷ֎ྗ߲f ͸࣍ͷΑ͏ʹͳΔɽamlωnf =−(7)θ̇ cos2 θ2k͜ΕΛࣜ (2) ʹ୅ೖ͠ɼ֎ྗͷৼΓࢠʹର͢Δ࢓ࣄ཰P Λ‫͢ࢉܭ‬Δͱɼ࣍ͷΑ͏ʹͳΔɽ
amlωn 2P = f θ̇ = −(8)θ̇ cos2 θ ≤ 02k͕ͨͬͯ͠ɼࣜ (5) ͷΑ͏ʹ୆ंมҐ͕ cn ؔ਺Ͱಈ25

間接フィードバック制御による振り子の振動制御（園部・東）͘Α͏ʹ͢Ε͹ɼ୆ं͸ৼΓࢠʹରͯ͠ θ = ±π/2 ʹ

2. 4໨ඪͱ͢Δ୆ंͷӡಈͷද‫ݱ‬͓͍ͯෛͷ࢓ࣄΛ༩͑ଓ͚Δɽ͢ͳΘͪɼৼΓࢠͷΤ

2. 2 અͰड़΂ͨΑ͏ʹɼ୆ंͷӡಈ͕ࣜ (5) Ͱ༩͑ωϧΪʔΛ‫ݮ‬গͤ͞ɼͦͷৼಈΛখ͘͢͞Δ͜ͱ͕ͰΒΕͨ৔߹ʹৼΓࢠͷৼಈΛখ͘͢͞Δ͜ͱ͕Ͱ͖Δɽ͖Δɽ

ࣜ (5) Ͱ༩͑Δ໨ඪͱ͢Δ୆ंมҐ x̄ Λࣜ (14) Ͱಋग़

2. 3 sn ؔ਺ͱ cn ؔ਺ͷಋग़લઅͰ͸ɼ୆ंมҐΛ cn ؔ਺Ͱ༩͑Δ͜ͱʹΑͬͯৼΓࢠͷৼಈΛ཈͑ΔޮՌ͕ಘΒΕΔ͜ͱΛࣔͨ͠ɽ͔͠͠ɼcn ؔ਺͸۩ମతʹ‫͢ࢉܭ‬Δ͜ͱ͕೉͍ͨ͠Ίɼ͜ΕΛ࣮‫͢ݱ‬ΔͨΊʹ͸ɼࣜ (5) ͷ୆ंมҐʹ cn ؔ਺ͨ͠ sin ϕ ͓Αͼ cos ϕ Λ࢖ͬͯද‫͢ݱ‬Δͱ࣍ͷΑ͏ʹͳΔɽx̄ = a cos(ϕ + ψ)(15)͜͜ʹɼa ͸໨ඪͱ͢Δ୆ंͷԟ෮ӡಈͷৼ෯Ͱ͋Γɼͷ୅ΘΓͱͳΔ۩ମతͳؔ਺Λ༩͑Δඞཁ͕͋Δɽͦψ ͸໨ඪʹର͢ΔมҐͷҐ૬஗ΕΛิਖ਼͢Δఆ਺Ͱ͋͜ͰɼྗֶతΤωϧΪʔ͕Ұఆͱ͍͏Ծఆͷ΋ͱͰɼΔɽࣜ (15) Λ࣌ؒඍ෼͢Δͱɼ໨ඪͱ͢Δ୆ंͷ଎౓sn ؔ਺͓Αͼ cn ؔ਺ΛৼΓࢠͷঢ়ଶྔ͔Βಋग़͢Δ͜ͱΛࢼΈΔɽͱՃ଎౓͸࣍ࣜͰಘΒΕΔɽθx̄˙ = −aωn cos sin(ϕ + ψ)(16)2
ࣜ (16) ͷಋग़ʹ͸ࣜ (11) ͷ 1 ֊࣌ؒඍ෼ͱࣜ (12) ͔ࣜ (3) ͷӡಈํఔࣜʹର͢ΔΤωϧΪʔอଘͷࣜ͸࣍ͷΑ͏ʹͳΔɽ
1(J + ml2 )θ̇2 + mgl(1 − cos θ)
2= mgl(1 − cos α)ࣜ (9) ΑΓɼ͕࣍ࣜಘΒΕΔɽθθ̇ = 2ωn k 2 − sin22(9)ΒಘΒΕΔ࣍ͷؔ܎ࣜθϕ̇ = ωn cos2Λ༻͍ͨɽ(17)Ҏ্ΑΓɼࣜ (15) ͓Αͼࣜ (16) Λຬͨ͢ԟ෮ӡಈ

(10)Λ୆ंʹͤ͞Δ͜ͱͰৼΓࢠͷৼಈ੍‫ޚ‬Λߦ͏ɽ·ͨɼପԁؔ਺ͷ฼਺͕ k = sin(α/2) (0 ≤ k ≤ 1) Ͱ͋Δ͜ͱΛߟྀ͢Δͱɼ࣍ͷΑ͏ʹ͓͘͜ͱ͕Ͱ͖Δɽθ
sin = k sin ϕ(11)2ࣜ (10) ͓Αͼࣜ (11) ΑΓɼ͕࣍ࣜಘΒΕΔɽ

3 . ୯ҰৼΓࢠϞσϧͷ੍ৼ੍‫ޚ‬ຊষͰ͸ਤ 1 ͷ୆ं‫ۦ‬ಈ‫ܕ‬୯ҰৼΓࢠϞσϧͷ੍ৼ੍‫ޚ‬ͷ৔߹ʹ͍ͭͯɼͦͷ੍‫ܥޚ‬ઃ‫݁ޚ੍ͱܭ‬Ռʹ͍ͭͯड़΂Δɽ

(12)

3. 1 ੍‫ܥޚ‬ઃ‫ܭ‬ຊ੍‫ޚ‬๏Ͱ͸ɼ௚઀ͷ੍‫ޚ‬ର৅ͱ͢Δͷ͸୆ंͷ

ࣜ (4) ͷୈ 2 ࣜͱࣜ (12) ΛൺֱΑΓɼྗֶతΤωϧΪʔΈͰ͋ΓɼৼΓࢠʹ͍ͭͯ͸໨ඪ஋ͱͯؒ͠઀తʹ͕Ұఆͱ͍͏৚݅Լʹ͓͍ͯ͸ cn(ωn t, k) = cos ϕ ͕ϑΟʔυόοΫΛߦ͏‫ͳͱܗ‬Δɽঢ়ଶํఔࣜ͸࣍ͷΑ੒Γཱͭ͜ͱ͕Θ͔Δɽ͏ʹͳΔɽθ̇ = 2ωn k cos ϕҎԼͰ͸ɼsn ؔ਺ʹ૬౰͢Δ sin ϕ ͱ cn ؔ਺ʹ૬
౰͢Δ cos ϕ Λঢ়ଶྔͷؔ਺ͱͯ͠ಋग़͢Δɽࣜ (10)ΑΓɼk ͸࣍ͷΑ͏ʹ‫ٻ‬ΊΒΕΔɽ
21θ
2ωn sink=+ θ˙22ωn2

A=(13)

ࣜ (12) ͔Βɼsin ϕ = sin(θ/2)/k ͱͳΔͷͰɼ͜Εʹࣜ (13) Λ୅ೖ͢Δ͜ͱͰ sin ϕ ͓Αͼ cos ϕ ͸࣍ͷΑ͏ʹ‫ٻ‬ΊΒΕΔɽ⎫⎪⎪⎪⎪⎪⎪
sin ϕ = ⎪⎪2⎪⎪θ⎬2ωn sin+ θ̇2 ⎪2⎪⎪⎪θ̇⎪⎪cos ϕ = ⎪⎪2⎪⎪θ⎪2ωn sin+ θ̇2 ⎪⎭2ẍ = Ax + Bu⎫⎪
⎪
⎬

0x ⎪
0 1, B= 2 , x=⎪ξẋ ⎭0 −2ξζ(18)

ࣜ (18) ʹରͯ͠ɼઢ‫ܗ‬ϑΟʔυόοΫ੍‫ޚ‬Λద༻͠ɼ‫ۃ‬഑ஔʹΑͬͯϑΟʔυόοΫήΠϯ k1 , k2 Λઃ‫͢ܭ‬Δɽ୆ंʹৼΓࢠΛ੍‫͢ޚ‬ΔͨΊͷӡಈΛ༩͑ΔͨΊʹɼ୆ंͷมҐͱ଎౓Λࣜ (15) ͓Αͼࣜ (16) ͔Β࣍ͷΑ͏ʹ༩͑Δɽθ
2ωn sin2

˙ − k2 (x − x̄)
u = −k1 (ẋ − x̄)

(14)

(19)

͜Ε͕ຊ੍‫ޚ‬๏ʹΑΔ௚઀ϑΟʔυόοΫͷ෦෼Ͱ͋˙ Λ௨ͯؒ͡઀ΓɼৼΓࢠ͔Βͷग़ྗ͸໨ඪ஋ (x̄, x̄)ϑΟʔυόοΫͱͯ͠࡞༻͢Δɽ

3. 2प೾਺Ԡ౴ʹ‫ ͮ͘ج‬ψ ͷઃ‫ܭ‬ࣜ (15) ͓Αͼࣜ (16) Ͱࣔͨ͠Ґ૬ͷิਖ਼͢ΔͨΊ

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弓削商船高等専門学校 紀要 第31号（平成21年）

ͷఆ਺ ψ Λۙࣅతʹ‫ٻ‬ΊΔɽࣜ (19) ͷ੍‫ޚ‬ೖྗΛ༩͑ͨ৔߹ʹɼx̄ ͱ x̄˙ ͕࣌ؒඍ෼ͷؔ܎Ͱ༩͑ΒΕΔ͜ͱͳΓɼθ = θ̇ = 0 ͱͳΔͱ͖ɼ୆ंͷ໨ඪӡಈ΋ࣗͱΛߟྀͯ͠ɼx̄ Λೖྗɼ୆ंมҐ x Λग़ྗͱߟ͑Δ

ࣜ (8) ͱಉ༷ʹ‫ٻ‬ΊΔͱɼͱঢ়ଶํఔࣜͱग़ྗํఔࣜ͸࣍ͷΑ͏ʹද͞ΕΔɽ⎫⎪ẋ = (A − Bk)x + Bkx̄⎪⎪⎪⎪⎬c = Cx
(20)
⎪⎪x̄
⎪
⎪
, C= 1 0 ⎪k = k1 k2 , x̄ =⎭x̄˙

a0 mlωn 2(25)θ̇ cos2 θ2ͱͳΔɽ͜Ε͸ɼৼΓࢠͷ֯଎౓͕࠷େͰ͋Γɼ͔ͭɼ୆ंӡಈʹΑͬͯੜ͡Δ‫׳‬ੑྗ͕ৼΓࢠͷճసӡಈʹ͜ͷೖग़ྗؔ܎Λ఻ୡؔ਺Λ༻͍ͯද͢ͱ࣍ͷΑ͏ʹ৐ʹൺྫ͢Δ͜ͱ͔ΒɼৼΓࢠͷྗֶతΤωϧΪʔʹͳΔɽ

Ԡͯ͡࢓ࣄ཰͕มԽ͢Δ͜ͱΛ͓ࣔͯ͠Γɼຊ੍‫ޚ‬ख

X = (G1 (s) + sG2 (s))X̄⎫⎬

G1 (s) G2 (s) = C(sI − A + Bk)−1 Bk⎭

વͱখ͘͞ͳΔɽ·ͨɼͦͷͱ͖ͷ୆ंͷ࢓ࣄ཰ P ΛP =−࠷΋Ө‫ڹ‬Λ༩͑Δ࠷Լ఺Ͱଟ͘ͷ࢓ࣄΛ͢Δ͜ͱΛද͓ͯ͠Γɼ͞ΒʹɼΤωϧΪʔ͕ৼΓࢠͷ֯଎౓ͷೋ๏͕ྗֶతͳ‫͔఺؍‬ΒΈͯଥ౰ੑͷߴ͍ख๏Ͱ͋Δ͜

(21)

͜͜ʹɼX, X̄ ͸ͦΕͧΕ x, x̄ Λϥϓϥεม‫ͨ͠׵‬΋ͷͰ͋Δɽೖྗ x̄, x̄˙ ͸ඍ෼ؔ܎Λҡ࣋ͨ͠··ࣜ (15)͓Αͼࣜ (16) Ͱ༩͑Δ͕ɼ͜͜Ͱ͸ۙࣅతʹҰఆ֯଎౓ ω ͷࡾ֯ؔ਺Ͱ༩͑ΒΕͨͱԾఆ͢Δͱɼप೾਺Ԡ౴ͷߟ͑ʹैͬͯೖग़ྗؒͷҐ૬஗ΕΛ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɽ੍ৼ੍‫࣌ޚ‬ͷৼΓࢠͷৼಈ਺͸‫ݻ‬༗ৼಈ਺ʹ΄΅౳͍͠ͱԾఆ͠ɼψ Λ͔࣍ࣜΒ‫ٻ‬ΊΔɽͱ͕Θ͔Δɽ

3. 4 ਺஋γϛϡϨʔγϣϯʹΑΔ‫ূݕ‬਺஋γϛϡϨʔγϣϯʹΑΔ‫ূݕ‬Λߦͬͨ݁Ռʹ͍ͭͯड़΂Δɽ·ͣɼ੍‫ޚ‬Ϟσϧͷ‫ܥ‬ύϥϝʔλΛද 1 ʹࣔ͢ɽ੍‫ܥޚ‬ઃ‫ܭ‬ύϥϝʔλʹ͍ͭͯ͸ɼ‫ۃ‬Λ (−10, −8)ͱ͠ɼࣜ (22) ͷ‫ࢉܭ‬ΑΓ ψ = 2.265 rad ͱͨ͠ɽॳ‫ظ‬֯౓Λ θ = 5π/6 rad ͱͯ͠਺஋γϛϡϨʔγϣϯΛߦͬͨɽa0 = 0.1 ͱͨ͠৔߹ͷ݁ՌΛਤ 3 ʹɼa0 = 0.2
ͱͨ͠৔߹ͷ݁ՌΛਤ 4 ʹࣔ͢ɽ

ψ = −∠(G1 (jωn ) + jωn G2 (jωn ))[G1 (jωn )] + ωn [G2 (jωn )]−1= − tan[G1 (jωn )] − ωn [G2 (jωn )]2 ͭͷ݁Ռͱ΋ʹɼৼΓࢠͷৼΕʹରͯ͋͠ΔҰఆͷҐ૬ࠩΛอͪͳ͕Βಉ‫ͯ͠ظ‬୆ं͕ಈ͍͓ͯΓɼ֤มҐͷॳ‫ظ‬஋͕େ͖͘ɼ‫͍ڧ‬ඇઢ‫ܗ‬ಛੑΛ༗͍ͯ͠Δ(22)ʹ΋ؔΘΒͣɼৼΓࢠͷৼಈΛ཈͑Δ੍‫͕ޚ‬༗ޮʹಇ͜͜ʹɼ[z] ͸ z ͷ࣮෦Λද͠ɼ[z] ͸ z ͷ‫ڏ‬෦Λද͍͍ͯΔ͜ͱ͕Θ͔Δɽ·ͨɼৼΓࢠͷৼಈΛ཈͑Δ͢ɽҎ্ͷํ๏ʹΑΓɼۙࣅతͰ͸͋Δ͕໨ඪ஋ʹର͜ͱͱ୆ंͷҐஔ੍‫͕ޚ‬ಉ࣌ʹ࣮‫͓͖ͯͰݱ‬Γɼͦͷ͢Δ୆ंͷҐ૬஗ΕΛิਖ਼͢Δ͜ͱ͕Ͱ͖Δɽ·ͨɼࡍͷ୆ंͷԟ෮ӡಈͷมҐ΋খ͘͞཈͑ΒΕ͍ͯΔɽຊۙࣅઃ‫ܭ‬ख๏ͷ༗ޮੑʹ͍ͭͯ͸ɼஶऀΒͷάϧʔҰ൪Լʹࣔͨ͠ପԁؔ਺ͷ฼਺ k ͸ৼΓࢠͷྗֶతΤϓʹΑΔଞͷ‫ڀݚ‬੒Ռ Ͱ‫͍ͯ͠ূݕʹط‬ΔͨΊɼ͜ωϧΪʔʹ૬౰͠ɼθ = π/2 ෇ۙͰ͸ৼΓࢠʹෛͷ࢓͜Ͱ͸লུ͢Δɽࣄ͕Ͱ͖ͳ͍ͨΊ‫ݮ‬গྔ͕খ͍͞΋ͷͷɼͦΕҎ֎ͷ

3. 3 ପԁؔ਺ͷ฼਺ʹ‫ͮ͘ج‬୆ंӡಈͷௐ੔্ड़ͷख๏Ͱ੍‫ޚ‬Λߦͬͨ৔߹ɼৼΓࢠͷৼಈ͕े෼খ͘͞ͳͬͨͱ͖ʹෆ۩߹͕ൃੜ͢ΔɽͦΕ͸ɼঢ়ଶͰ͸࣌ؒͷ‫ܦ‬աͱ‫ݮʹڞ‬গ͍ͯ͠Δɽࣜ (25) Ͱࣔͨ͠ৼΓࢠͷ࢓ࣄ཰͸ΤωϧΪʔʹൺྫ͢Δ΋ͷͰࣜ (14) Ͱ‫ٻ‬ΊΒΕΔ sin ϕ, cos ϕ ͷ෼฼ͱ෼ࢠ͕‫ʹڞ‬͍Δɽ

[7]

͋ΔͨΊɼk ͷ࣌ؒมԽ͸ࢦ਺ؔ਺తͳ΋ͷͱͳͬͯΓɼ݁Ռͱͯ͠νϟλϦϯάͷΑ͏ͳ‫ڍ‬ಈͱͳΔ͜ͱ

a0 = 0.1 ͱ a0 = 0.2 ͷ৔߹Λൺֱ͢Δͱɼ‫ऀޙ‬ͷ΄͏͕ऩଋ͕࣌ؒ໌Β͔ʹ଎͍͜ͱ͔Βɼa0 ͷઃఆͰ͋Δɽ͜ΕΛ๷͙ͨΊʹɼࣜ (15) ͓Αͼࣜ (16) ͷ஋ʹΑͬͯৼಈͷऩଋੑೳΛมԽͤ͞Δ͜ͱ͕Ͱ͖Δ

θϩʹۙ͘ͳΔͨΊɼ୆ंͷ໨ඪӡಈ͕ෆ࿈ଓతͱͳ୆ंͷ໨ඪӡಈͷ܎਺ a Λ࣍ͷΑ͏ʹ༩͑Δɽ

a = a0 k(23)͜ͱ͕Θ͔Δɽ͜Ε͸ɼa0 ͕ৼΓࢠͷऩଋ଎౓ͱ୆ंมҐͷେ͖͞ͱ͍͏τϨʔυΦϑͷؔ܎Λઃఆ͢ΔύϥϝʔλͰ͋Δ͜ͱΛද͍ͯ͠Δɽ͜͜ʹɼk ͸ପԁؔ਺ͷ฼਺Ͱ͋Γɼࣜ (13) ΑΓ‫ٻ‬ΊΒҎ্ΑΓɼຊ੍‫ޚ‬ख๏ʹΑͬͯৼΓࢠͷৼಈΛ੍‫ޚ‬ΕΔɽ͜ΕʹΑͬͯɼԾʹ୆ंͷ໨ඪ஋Λ x̄ = a cos ϕՄೳͰ͋Δͱಉ࣌ʹɼͦͷ‫ڍ‬ಈ΍੍‫ޚ‬ύϥϝʔλͷઃͰ༩͑ͨ৔߹ʹ͸ɼ‫ܭ‬ख๏͕ྗֶత‫͔఺؍‬Βଥ౰Ͱ͋Δ͜ͱ͕ࣔ͞Εͨɽx̄ = a0 k cos ϕ =a0θ̇2ωn(24)

27

間接フィードバック制御による振り子の振動制御（園部・東）4 . ݁

Table 1 Values of system parametersm0.0499
kglJ
c

0.20
6.53 × 10−4
2.36 × 10−4ξζ

6.0
3.5

‫ݴ‬୆ं‫ۦ‬ಈ‫ৼܕ‬ΓࢠϞσϧʹର͠ɼପԁؔ਺ʹ‫ͮ͘ج‬

mkg · m2
Nms/radԟ෮ӡಈΛ୆ंʹ༩͑Δ͜ͱͰৼΓࢠͷৼಈΛ੍‫͢ޚ‬Δख๏ΛఏҊͨ͠ɽಘΒΕͨ݁࿦͸࣍ͷ௨ΓͰ͋Δɽɾ ৼΓࢠΛ௚઀ͷ੍‫ޚ‬ର৅ͱͤͣɼ୆ंͷΈʹର੍ͯ͠‫ܥޚ‬ઃ‫ܭ‬Λߦ͍ɼৼΓࢠͷৼಈ੍‫ޚ‬͸୆ंӡಈͷ໨ඪ஋ͱ͍͏ؒ઀੍‫ͯ͠ͱޚ‬༩͑ͨɽ͜ΕʹΑͬͯɼ୆ंͷҐஔ੍‫ৼͱޚ‬Γࢠͷৼಈ੍‫͍ͱޚ‬T (rad)2͏ 2 ͭͷ੍‫ޚ‬Λಉ࣌ʹແཧͳ࣮͘‫ͨ͠ݱ‬ɽ0ɾ ୆ंͷҐஔ੍‫ৼͱޚ‬Γࢠͷৼಈ੍‫ޚ‬ͷॏΈ͚ͮʹ

–2͍ͭͯ͸ɼ୆ंͷԟ෮ӡಈͷύϥϝʔλ a0 ͷΈx (m)0.2ʹΑͬͯௐ੔ՄೳͰ͋Δɽɾ ୆ंͷԟ෮ӡಈͷେ͖͞Λପԁؔ਺ͷ฼਺ k ʹ߹

–0.2ΘͤͯมԽͤ͞Δ͜ͱͰৼΓࢠͷΤωϧΪʔঢ়ଶu (V)01ʹԠͯ͡ৼಈΛ‫ݮ‬গͤ͞Δ͜ͱ͕Մೳͱͨ͠ɽ0ࠓ‫ޙ‬͸ɼ࣮‫͍͓ͯʹػݧ‬ຊ੍‫ޚ‬๏ͷ༗ޮੑͷ֬ೝΛߦ͏ͱͱ΋ʹɼຊ੍‫ޚ‬๏Λߦͬͨࡍͷ‫ػ‬ցతෛՙͷܰ

–1‫ݕ͍ͯͭʹݮ‬౼͍ͨ͠ͱߟ͍͑ͯΔɽ

1k

จ
0024

Time (s)ΕࢭΊ੍‫΁ޚ‬ͷద༻ʵ, ‫ܭ‬ଌࣗಈ੍‫ֶޚ‬ձ࿦จू,T (rad)2Vol.41-6, pp.509-517, 2005.

0[3] ҏ౻ ༸ɾླ໦ ߒฏ, ఱҪΫϨʔϯͷՙ༳Ε཈੍ख๏ʹؔ͢Δ‫ڀݚ‬ʢ࿈੒ৼಈΛߟྀ͠ՙ༳ΕΛ–20.2x (m)[1] ࡔ࿨ Ѫ޾, ఱҪ૸ߦΫϨʔϯͷϞσϦϯάͱ࠷ద੍‫ޚ‬, ‫ܭ‬ଌࣗಈ੍‫ֶޚ‬ձ࿦จू, Vol.34-8, pp.962967, 1998.
[2] খ୩ ੪೭ɾେ಺ ໜਓɾ݀ਧ խහɾཱུ ߁ࢤ, H2 ίϯτϩʔϥͷ௿࣍‫ݩ‬Խઃ‫ܭ‬ʵ૸ߦΫϨʔϯͷৼ

Fig. 3 Simulation result (a0 = 0.1)཈੍ͨ͠ൖૹख๏ʣ, ೔ຊ‫ػ‬ցֶձ࿦จू (C ฤ),0Vol.72-720, pp.2406-2412, 2006.
[4] ࡾ޷ ޹యɾࣉౢ Ұ඙ɾ৿ా ਖ਼ଇ, ࣌มৼಈ‫ߏػ‬ʹର͢Δ੍ৼϑΟʔυϑΥϫʔυ੍‫ޚ‬ೖྗͷಋग़ͱΫϨʔϯ΁ͷԠ༻, ೔ຊ‫ػ‬ցֶձ࿦จू (C ฤ),
Vol.64-624, pp.2859-2866, 19986.

–0.2
1u (V)‫ݙ‬

0–1[5] K. J. Åström and K. Furuta, Swing up a pendulum
by energy control, Automatica, Vol.36-2, pp.287295, 2000.

k1[6] ‫ాށ‬, ପԁؔ਺ೖ໳, ೔ຊධ࿦ࣾ, 2001.
0024Time (s)Fig. 4 Simulation result (a0 = 0.2)

[7] Ԃ෦ ‫߁ݩ‬ɾদ࡚ ݈Ұ࿠ɾۙ౻ ޹޿ɾफ࿨ ৳ߦ,Ϟʔυղੳʹ‫ͮ͘ج‬෦෼੍ۭؒ‫ޚ‬๏ʹؔ͢Δ‫ڀݚ‬(ୈ 2 ใɼඇઢ‫ৼڞܗ‬Λར༻ͨ͠ฒྻೋॏ౗ཱৼࢠͷৼΓ্੍͛‫)ޚ‬, ೔ຊ‫ػ‬ցֶձ࿦จू (C ฤ),Vol.74-745, pp.2198-2205, 2008.